Diffeomorphisms with Banach space domains

TL;DR

This paper investigates conditions under which a locally diffeomorphic map between Banach spaces is globally injective or bijective, using scalar coercive functions to establish sufficient and necessary criteria.

Contribution

It provides new sufficient conditions for global injectivity and bijectivity of $C^1$ maps between Banach spaces, extending classical results to infinite-dimensional settings.

Findings

Sufficient conditions for a local diffeomorphism to be globally injective.

Necessary and sufficient conditions for bijectivity when domain and codomain are $ ^n$.

Use of scalar coercive functions as key tools in establishing global diffeomorphism criteria.

Abstract

Our basic element is a $C^1$ mapping $f:X\to Y$, with $X,Y$ Banach spaces, and with derivative everywhere invertible. So $f$ is a local diffeomorphism at every point. The aim of this paper is to find a sufficient condition for $f$ to be injective, and so a global diffeomorphism $X\to f(X)$, and a sufficient condition for $f$ to be bijective and so a global diffeomorphism onto $Y$. This last condition is also necessary in the particular case $X=Y=\R^n$. In these theorems the key role is played by nonnegative auxiliary scalar coercive functions.